Asymptotics of the expected posterior

Bertrand Clarke, Dongchu Sun

Research output: Contribution to journalArticle

5 Scopus citations

Abstract

Let (X1, . . . , Xn) be independently and identically distributed observations from an exponential family pθ equipped with a smooth prior density w on a real d-dimensional parameter θ. We give conditions under which the expected value of the posterior density evaluated at the true value of the parameter, θ0, admits an asymptotic expansion in terms of the Fisher information I(θ0), the prior w, and their first two derivatives. The leading term of the expansion is of the form nd/2C1(d, θ0) and the second order term is of the form nd/2-1c2(d, θ0, w), with an error term that is o(nd/2-1). We identify the functions C1 and C2 explicitly. A modification of the proof of this expansion gives an analogous result for the expectation of the square of the posterior evaluated at θ0. As a consequence we can give a confidence band for the expected posterior, and we suggest a frequentist refinement for Bayesian testing.

Original languageEnglish (US)
Pages (from-to)163-185
Number of pages23
JournalAnnals of the Institute of Statistical Mathematics
Volume51
Issue number1
DOIs
StatePublished - Jan 1 1999

Keywords

  • Asymptotics
  • Bayes factor
  • Chi-squared distance
  • Expected posterior
  • Relative entropy

ASJC Scopus subject areas

  • Statistics and Probability

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